Show that $S$ is compact.

Question

Let $A \in M_n(\bf C)$ and let $S=\{ A \in M_n(\bf C) : A=A ^*$ and $\rho (A) \leq 1 \}$, $\rho(A)$ denotes spectral radius of $A$.

Show that $S$ is compact subset of $M_n(\bf C) \cong \bf C^{n^2} $ with usual topology.

$S$ is subset of $\bf O_n(C)$ which is compact so I think its enough to show that $S$ is closed.Please help.

Answer 1

For the hermitian matrices we have $||A||= \rho(A)$ and $\rho$ is given bounded , thus the set is bounded and since $\rho$ is a continuous map, inverse image of $(-\infty,1]$ is closed and thus your set is closed too. So it is compact.